11
Multi-Valued Functionals, One-Forms
and Deformed de Rham Complex
∗
D.V. Millionschikov
Summary.We discuss some applications of the Morse–Novikov theory to some
problems in modern physics, which appears as a non-exact closed 1-formω(multi-
valued functional). We focus our attention mainly on the cohomologyHλω∗(Mn,R)
of the de Rham complexΛ∗(Mn) of a compact manifoldMnwith a deformed
differential dω=d+λω. Using Witten’s approach to the Morse theory one can
estimate the number of critical points ofωin terms ofH∗λω(Mn,R) with sufficiently
large values ofλ(torsion-free Novikov’s inequalities).
We show that for an interesting class of solvmanifoldsG/Γ the cohomology
Hλω∗(G/Γ,R) can be computed as the cohomologyHλω∗(g) of the corresponding
Lie algebragassociated with the one-dimensional representationρλω. Moreover
Hλω∗(G/Γ,R) is almost always trivial except a finite number of classes [λω]in
H^1 (G/Γ,R).
12.1 Introduction
In the beginning of the 1980s Novikov constructed [1, 2] an analogue of the
Morse theory for smooth multi-valued functions, i.e. smooth closed 1-forms on
a compact smooth manifoldM. In particular he introduced the Morse-type
inequalities (Novikov’s inequalities) for numbersmp(ω) of zeros of indexpof
a Morse 1-formω.
In [3, 4] a method of obtaining the torsion-free Novikov inequalities in
terms of the de Rham complex of manifold was proposed. This method was
based on Witten’s approach [5] to the Morse theory. Pazhitnov generalized
Witten’s deformation d +tdf(fis a Morse function onM) of the standard
differential d inΛ∗(M) by replacing dfby an arbitrary Morse 1-form onM.
For sufficiently large real valuestone has the following estimate (torsion-free
Novikov inequality [4, 6]):
mp(ω)≥dimHtωp(M,R),
∗Partially supported by the Russian Foundation for Fundamental Research, grant
no. 99-01-00090 and PAI-RUSSIE, dossier no. 04495UL